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MathGroup Archive 2002

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A Bessel integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36779] A Bessel integral
  • From: Roberto Brambilla <rlbrambilla at cesi.it>
  • Date: Wed, 25 Sep 2002 01:50:58 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hi to all Mathematica friend!

I am considering the following integral

W[m_,n_]:=Integrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]

where m,n are reals >=0. With Mathematica 4.1 I obtain:

If[Re[m+n]>-1, -Cos[(m-n)Pi/2]/(2 Pi)*
(2 EulerGamma + Log[4] + 
PolyGamma[0, 1/2(1 + m - n)] + 
PolyGamma[0, 1/2(1 - m + n)] + 
2PolyGamma[0, 1/2(1 + m + n)])

and so using this answer as a definition I obtain  

W[0,0]=-(2 EulerGamma + Log[4] + 4 PolyGamma[0, 1/2])/(2 Pi)=0.84564

I suspect that these integrals are divergent (*). So I try the numerical
integration:

NW[m_,n_]:=NIntegrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]

so that

NW[0,0]=11.167

Othe couples are

W[1,0]=Indeterminate  NW[1,0]=0.597973

W[0,1.5]=0.537095     NW[0,1.5]=-5.79306

W[1,1]=0.20902        NW[1,1]=17.5425

W[2,0]=0.427599       NW[2,0]=-6.83464

W[2,1]=Indeterminate  NW[2,1]=4.69013

(*) The integral is a particular case of the Weber-Schafheitlin integrals
(Abramowitz, 11.4.33).

Any explanation about the analytical expression will be gratefully accepteed.

Roberto.

Roberto Brambilla
CESI
Via Rubattino 54
20134 Milano
tel +39.02.2125.5875
fax +39.02.2125.5492
rlbrambilla at cesi.it



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